Detection and correction of spectral and spatial
misregistrations for hyperspectral data using phasecorrelation method
Naoto Yokoya,1,∗ Norihide Miyamura,2 and Akira Iwasaki2
1
2

Department of Aeronautics and Astronautics, University of Tokyo, Japan

Research Center for Advanced Science and Technology, University of Tokyo, Japan∗

Corresponding author: yokoya@sal.rcast.u-tokyo.ac.jp
Abstract

Hyperspectral imaging sensors suﬀer from spectral and spatial misregistrations.
These artifacts prevent the accurate acquisition of spectra and thus reduce classification accuracy. The main objective of this work is to detect and correct spectral andspatial misregistrations of hyperspectral images. The Hyperion visible near-infrared(VNIR) subsystem is used as an example. An image registration method based onphase correlation demonstrates the precise detection of the spectral and spatial misregistrations. Cubic spline interpolation using estimated properties makes it possible
to modify the spectral signatures. The accuracy of the proposed postlaunch estimationof the Hyperion characteristics is comparable to that of the prelaunch measurements,
which enables the precise onboard calibration of hyperspectral sensors.

1.

Introduction

Hyperspectral imaging sensors enable high-spectral-resolution remote sensing, such as theprecise mapping of diﬀerent types of vegetation and minerals. These sensors collect spectrograms as sets of data cubes, each of which represents a spectral band radiance. These images
form a three-dimensional (3D) cube, composed of cross-track (x ), along-track (y), and spectrum (λ) dimensions. Hyperspectral data contains hundreds to thousands of spectral bands.
Owing to the high-resolution spectral profile in each pixel, it is possible to discriminate
among land-cover classes that are spectrally similar [1].
Hyperspectral imaging sensors suﬀer from spectral and spatial misregistrations. These
artifacts are caused by optical aberrations and misalignments in pushbroom systems, wherecross-track and spectral pixels are continuously recorded at the same time using a twodimensional (2D) detector array. Spectral misregistration, also known as a gsmileh or gfrownh
1

curve, is a shift in wavelength in the spectral domain, which is a function of the cross-track
pixel number. Spatial misregistration, also called gkeystoneh, corresponds to band-to-bandmisregistration. Both types of misregistration distort the spectral features and thus reduceclassification accuracy. Past studies showed that very high sensor accuracy is required toobtain spectral signatures useful for scientific research [2, 3]. A spectral uncertainty of lessthan 1% of the full width at half maximum (FWHM) throughput of the spectral response
function is necessary [2]. Furthermore, the maximum spatial misregistration has to be lessthan 5% of the pixel size [3]. Hereafter, we refer to spectral and spatial misregistrations assmile and keystone, and denote them as λ(x) and x(λ), respectively.
The Hyperion is a hyperspectral imager onboard the Earth Observing 1 (EO-1) satellite
launched on November 21, 2000. The Hyperion system, consisting of visible near-infrared(VNIR) and shortwave infrared (SWIR) subsystems, acquires data in 198 spectral bands.
The Hyperion VNIR spectrometer obtains optical images in 50 bands over a range of 426.8925.4nm and the FWHM is 10.2nm. In this work, the Hyperion VNIR subsystem is used as
an example because the signal-to-noise ratio (SNR) is high and the number of defects is low
as a space-borne hyperspectral sensor.
The prelaunch smile property of Hyperion was characterized by the developer using a
multispectral test bed and was attached to the Hyperion dataset [4]. The maximum smileof VNIR is about 20-30% of FWHM. Most researchers use prelaunch information to correct
spectral misregistration. However, a number of previous results indicate that there is a diﬀerence in the smile property before and after the launch [4, 5]. The postlaunch smile property
was detected by correlating the sensor-measured spectrum with a spectrum modeled using
atmospheric absorption lines. The estimation accuracy and robustness for spatially and temporally diﬀerent scenes have not evaluated with suﬃcient accuracy. Some smile correction
methods have been proposed but they failed to correct the smile eﬀect entirely [6].
The prelaunch keystone property was measured at 20 locations using a point source [7].
The maximum keystone of VNIR is over 10% of the pixel size. The postlaunch keystone
was also measured by the developer using lunar images obtained in orbit. However, thisinformation is limited by the cross-track numbers and is not attached to the Hyperion data.
Therefore, almost all users use the data without correcting the spatial misregistration. In paststudies on keystone detection was proposed, a scene-based method based on edge detection
using a sharpening filter [8, 9]; however, the result was diﬀerent from prelaunch information
after the launch.
Since the characteristics of a space-borne hyperspectral sensor may change after a launch,
precise onboard calibration is an important and challenging issue. The main objective of thiswork is to detect and correct the spectral and spatial misregistrations only using the observed
data. Two subpixel image registration methods based on normalized cross-correlation (NCC)

2and phase correlation (PC) are used to detect the artifacts, and their eﬀectiveness is evaluatedfrom the viewpoints of accuracy and robustness. Cubic spline interpolation is adopted tocorrect the artifacts because of its good trade-oﬀ between smoothness and shape preservation[10]. Methods of evaluating hyperspectral sensor characteristics are also described.
2.

Subpixel Image Registration Methods

Subpixel image registration methods can detect smile and keystone properties. Smile is detected by estimating the distortion of the atmospheric absorption line in the spectrum image,and keystone is detected by estimating the band-to-band misregistration of all subscenes.
Two diﬀerent subpixel estimation methods are introduced as follows.
2.A.

Normalized Cross-Correlation Method

The normalized cross-correlation function between the source and destination subscenes iscalculated by the following equation:∑ij {f (i, j) − f } × {g(i − m, j − n) − g},C(m, n) = ∑(1)∑( ij {f (i, j) − f }2 × ij {g(i − m, j − n) − g}2 )1/2
where f (i, j) and g(i, j) denote pixel values of the source and destination subscenes of (i, j),∑∑ 1 ∑N2
respectively. ij denotes N
i=1
j=1 , and (N1 , N2 ) is the size of the correlation window. The
misregistration can be estimated by the following steps. First, the integer set (m0 , n0 ) withthe maximum value of correlation is selected as the candidate for the misregistration between
two images. Next, the point with highest correlation is obtained at a subpixel level by fitting
a parabola to three neighboring correlation values. (δ1 , δ2 ), the subpixel misregistration of
two directions, is given by((δ1 , δ2 ) =2.B.

C(m0 − 1, n0 ) − C(m0 + 1, n0 ),2C(m0 − 1, n0 ) − 4C(m0 , n0 ) + 2C(m0 + 1, n0 ))C(m0 , n0 − 1) − C(m0 , n0 + 1)
. (2)2C(m0 , n0 − 1) − 4C(m0 , n0 ) + 2C(m0 , n0 + 1)

Phase Correlation Method

High-accuracy image registration techniques based on PC have been developed [11-14]. For
simplicity, we employ PC-based image registration using fitting with the one-dimensional(1D) sinc function, which directly estimates the misregistration [14]. By using a large numbers of subscenes to estimate each misregistration, robust performance and high subpixel
estimation accuracy are achieved.
We consider two N1 × N2 images, f (n1 , n2 ) and g(n1 , n2 ), with index ranges of n1 =−M1 , ..., M1 and n2 = −M2 , ..., M2 for mathematical simplicity. F (ω1 , ω2 ) and G(ω1 , ω2 )3

denote the 2D discrete Fourier transforms (2D DFTs) of the two images. The cross-phase
spectrum R(ω1 , ω2 ) is defined as
F (ω1 , ω2 )G∗ (ω1 , ω2 )R(ω1 , ω2 ) =.
| F (ω1 , ω2 )G∗ (ω1 , ω2 ) |(3)Here, G∗ is the complex conjugate of G.
Let fc (x1 , x2 ) and fc (x1 − δ1 , x2 − δ2 ) be two images defined in continuous space that diﬀer
only by a displacement (δ1 , δ2 ). When f (n1 , n2 ) and g(n1 , n2 ) are assumed to be sampled
images of fc (x1 , x2 ) and fc (x1 − δ1 , x2 − δ2 ), respectively, according to the Fourier shift
theorem, their corresponding 2D DFTs, F (ω1 , ω2 ) and G(ω1 , ω2 ), are related byG(ω1 , ω2 ) ≃ F (ω1 , ω2 ) · e

−j2π

(

ω1
ω
δ + 2δ
N1 1 N2 2

)

.

(4)

The phase correlation function r(n1 , n2 ) is defined as the 2D inverse discrete Fourier transform (2D IDFT) of R(ω1 , ω2 ). Thus, the phase correlation r(n1 , n2 ) is given byr(n1 , n2 ) = F −1 [R(ω1 , ω2 )] ≃ α ·sinπ(n1 + δ1 ) sinπ(n2 + δ2 ),π(n1 + δ1 )π(n2 + δ2 )(5)

where α ≤ 1. The peak position of the phase correlation function corresponds to the misregistration between two images. The important procedures for high-accuracy subpixel image
registration are listed below.
(i) Windowing to reduce boundary eﬀects
Because of the periodicity of the DFT, discontinuities occur at every edge in the 2D DFT
computation. Applying a 2D Hanning window to the input images makes it possible to reducethese boundary eﬀects. The 2D Hanning window is defined by
( )
( )
πn2
1
1 + cos πn
1
+
cos
M1
M2
w(n1 , n2 ) =
.
(6)
2
2
(ii) Modifying cross-phase spectrum using weighting function
Natural images may have a low SNR in the high-frequency components compared with in the
low-frequency components. Applying a low-pass weighting function H(ω1 , ω2 ) to R(ω1 , ω2 )and eliminating the high-frequency components improve the subpixel estimation accuracy.
The simplest weighting function is given by1 | ω |≤ U , | ω |≤ U
1
1
2
2
,
(7)
H(ω1 , ω2 ) =0 otherwise
where U1 and U2 are integers satisfying 0 ≤ U1 ≤ M1 and 0 ≤ U2 ≤ M2 . R(ω1 , ω2 ) is multiplied by H(ω1 , ω2 ) when calculating the 2D IDFT. Then the modified r(n1 , n2 ) is obtained4

byr(n1 , n2 ) ≃ α ·sin NV11 π(n1 + δ1 ) sin NV22 π(n2 + δ2 )π(n1 + δ1 )π(n2 + δ2 ),(8)

where V1 = 2U1 + 1 and V2 = 2U2 + 1.
(iii) Fitting with 1D sinc function
The location of the peak correlation can be directly estimated with 1/10-1/100-pixel accuracy
by fitting the sinc function to the calculated data array of r(n1 , n2 ) [14]. The misregistrations(δ1 , δ2 ) and the value α can be obtained independently, assuming that one misregistration isfixed. Thus, for mathematical simplicity, a 1D version of the correlation function is consideredin the following description. The 1D correlation function is given byr(n) ≃ β ·Vπ(n + δ)sin N,
π(n + δ)(9)where β ≤ 1, n = −M, ..., M is the index range, N = 2M + 1 is the signal length, δ is themisregistration, and V is the size of the low-pass weighting function. The misregistration
can be estimated by the following steps. First, the integer n0 with the maximum value ofcorrelation is selected as the candidate for the misregistration between two signals. Next, thepoint with the highest correlation is obtained at a subpixel level by fitting the sinc function
to three neighboring correlation values. The value of δ is given by
δ=
3.
3.A.

r(n0 − 1) − r(n0 + 1)
( V)
− n0 .
r(n0 ) + r(n0 + 1)
r(n0 − 1) − 2cos π N(10)Results and Discussion
Smile Property

3.A.1. Smile Indicator

Spectral smile becomes observable when the image is transformed into maximum noise fraction (MNF) space [15]. The MNF transformation produces new components ordered by image
quality by projecting the original image into a space where the new components are sorted
in order of SNR. The most important procedure in MNF is to estimate the noise covariancematrix eﬀectively. To obtain the noise covariance matrix, we estimate the signal part of eachpixel. There are many models that can be used to estimate the value of ŝx,y,λ , the signal partof the original value sx,y,λ . To visualize smile clearly in low-order MNF space, we adopt
as(x < W )x+1,y,λ(1 ≤ y ≤ H, 1 ≤ λ ≤ N ).
(11)ŝx,y,λ =asx−1,y,λ (x = W )Here, W and H are the width and height of the image, respectively, and N is the total
number of bands. The residual r = s − ŝ is the noise value of each pixel. The coeﬃcient a iscomputed to minimize Σr2 .
5

For Hyperion images with significant smile, a brightness gradient appears in the first eigenvalue image, MNF1. Fig. 1 shows MNF1 obtained from bands 8 (426.8nm) to 57 (925.4nm)
for a desert scene in Chile, the spectral profile of the scene, and the weighting factors that
contribute to MNF1. In this case, MNF1 is mainly dominated by bands 40 (752.4nm) and 42(772.8nm), near the oxygen absorption band 41 (762.6nm). This means that bands 40 and
42 are strongly aﬀected by the smile eﬀect because of the distorted oxygen absorption line.
Thus, we use the radiance diﬀerence between bands 40 and 42 as an indicator of spectral
smile.
3.A.2. Normalized Cross-Correlation Method

We detect the smile property λ(x) of band 41 by estimating the distortion of the oxygen
absorption line in the spectrum image, or the x -λ plane, using the subpixel image registration
method based on NCC. When a scene is assumed to be physically homogeneous at eachalong-track position, the subpixel image registration method based on applying NCC to thisline in the cross-track direction enables the detection of spectral distortion as a functionof the cross-track number. According to the prelaunch smile property of Hyperion VNIR,
smile λ(x) is thought to be quadratic function of x; thus, we finally estimate spectral smileusing a quadratic approximation of the average spectral distortion. The desert scene inChile observed on December 25, 2002, is used as a homogeneous scene. The size of thecorrelation window (Nx , Nλ ) is set as (1, 7) with band 41 at the center. When Nλ increases,
the detected distortion becomes increasingly aﬀected by other bands, and when Nλ decreases,
subpixel estimation accuracy deteriorates. The detector of Hyperion VNIR is composed offour units. The SNR of the quad 2D detector array with cross-track numbers of 129-256
and spectral channels 36-70 is higher than that of the other three quads [4]. The cross-tracknumber of the source subscene is 129 owing to the high SNR and the intermediate cross-tracknumber. Before the subpixel image registration process, the NCC coeﬃcient is calculated in acorrelation window between the source spectrum and the destination spectrum. Along-track
positions with uniform spectral response are selected by comparison with the cross-track
average threshold, NCC > 0.9.
The detected distortion of the oxygen absorption line (band 41) is shown in Fig. 2(a). This
indicator shows only the relative spectral distortion. The estimated smile property of band
41, adjusted to make the average spectral misregistration equal to zero, is shown in Fig. 3,
and it is obviously diﬀerent from the prelaunch property. We correct bands 40 and 42 using
cubic spline interpolation with the estimated spectral property and evaluate the validityof the correction by considering the radiance diﬀerence between bands 40 and 42, both ofwhich are strongly aﬀected by the smile eﬀect. Fig. 4 shows the radiance diﬀerence between
bands 40 and 42 and Fig. 5 shows profiles of the radiance diﬀerence. In the original data, the6brightness gradient due to smile can be seen clearly. In the data corrected using the prelaunch
spectral property, a reverse brightness gradient appears because of overcorrection. In the data
corrected using the spectral property estimated by the NCC method, the brightness gradientstill remains. This means that the spectral distortion estimated by the NCC method is smaller
than the true distortion. That may be because of the pixel-locking eﬀect, the phenomenon
that the estimated subpixel values tend to be biased toward integer values depending on theimage characteristics and fitting function [16]. We examined this phenomena by obtaining
simulation data with the applied smile property. To solve this problem, an ad hoc analysis inwhich the pixel-locking eﬀect is calculated is needed, but such an analysis is time-consuming
and reduces the robustness to spectral feature of scenes.
3.A.3. Phase Correlation Method

Similarly to in the NCC method, the smile property of band 41 is estimated by applying
the subpixel image registration method based on PC to the oxygen absorption line. We usedfour scenes in the estimation, two from the Chile desert and two of Cuprite. The cross-tracknumber of the source subscene and the physically homogeneous area selection are the sameas those in the NCC method. The size of the correlation window is set as (Nx , Nλ ) = (1, 11),
with band 40 at the center. After investigating the subpixel estimation accuracy of thephase correlation method using 1D sample signals, it is found that the estimation accuracy
significantly deteriorates when the length of the 1D signal is less than 11. This length isgreater than that for NCC because a Hanning window is applied. The oxygen absorption
line is the dominant signal in each subscene. Thus, the calculated spectral misregistration
can be regarded as a spectral property of band 41. The size of the weighting function H,
which modifies the cross-phase spectrum, is set as V = 3.
In Figs. 2(b) and 3, the detected distortion and the smile property estimated by the PC
method, examined for the same scenes as those in the NCC method, are illustrated. The
results are similar to the prelaunch property, although there are some diﬀerences. Figs. 4
and 5 demonstrate that the brightness gradient disappears in the corrected data when using
the smile property estimated by the PC method. This evaluation of smile was more accurate
than that using the prelaunch property. The smile property should have changed after thelaunch, and that of band 41 detected by the PC method has been proved to be valid by MNF
analysis and from the band diﬀerence. Some optical components may have been rotated ordisplaced owing to mechanical vibration during the launch and the sensors may have decayedover time. In the optical model, the smile property is assumed to be a quadratic curve [17],and we use the approximation λ(x) = a(x − x0 )2 + b. In Table 1, the quadratic coeﬃcient a,
the cross-track axis x0 , and the residual sum of squares (RSS) are listed for four test scenes.
Similar results are obtained from spatially and temporally diﬀerent scenes. The maximum

7

diﬀerence between the estimated smile properties is about 0.01 pixel. In summary, the PC
method is suitable for precise and robust onboard spectral smile calibration.
3.B.

Keystone Property

In the detection of the smile property of band 41, it was suggested that the PC method isfree from the pixel-locking eﬀect and superior to the NCC method. Thus, the PC method
is used to detect the keystone property. A subpixel image registration method based on PC
can estimate spatial misregistration in any physically homogeneous subscene. We selected
over 10,000 subscenes on an image, or the x-y plane. The size of the subscene was (Nx , Ny ) =(31, 31) because of the high subpixel estimation accuracy and robust performance. Band 31(660.9nm) was chosen as the reference owing to its central position in the Hyperion VNIR
spectral range on the x -λ plane and its relatively high SNR. Subscenes with uniform spectral
responses are selected by comparison with the band-to-band average threshold, NCC > 0.9.
V , the size of the weighting function, is set to 5. We investigated eleven scenes, which included
various types of terrain such as desert, mountains, and urban areas. The ocean and otherlow-radiance subscenes are inappropriate for keystone detection owing to the low SNR andlack of characteristic imagery.
Fig. 6 shows the detected spatial misregistration examined using the desert scene in Chileobserved on December 25, 2002, the same scene as that used for smile detection. Although
all plots exhibit large variations due to the low SNR of the sensor, they indicate a linear
tendency of the estimated keystone for each cross-track number. This is probably because
rotational misalignment was the dominant cause of keystone. For each cross-track number,the keystone property is approximated as a linear function. The standard deviation of theslopes is 3.38% of the average slope, as shown in Fig. 6. Thus, the spatial keystone property isthought to be almost the same for each cross-track number. Finally, we obtained x(λ) = cλ+dusing a linear approximation for all subscenes, which is consistent with the prelaunch andpostlaunch spatial misregistration measured by the developer [7].
Fig. 7 shows the keystone before and after keystone correction by spline interpolation.
The disappearance of keystone after the correction shows the validity of the estimation.
Fig. 8 shows the spatial misregistrations between bands 13 (477.7nm) and 50 (854.2nm)
estimated from the eleven scenes. The standard deviation is 9.18 × 10−3 pixels. Since thekeystone property does not significantly change in the optical model, the similar band-toband misregistrations demonstrate the accuracy of the PC method for the various spatiallyand temporally diﬀerent scenes. The keystone detection method based on PC has been proved
to be suitable for precise and robust onboard keystone calibration.

8

4.

Conclusion

In this paper we proposed a method for estimating postlaunch hyperspectral sensor characteristics. The image registration method based on phase correlation can detect spectral andspatial misregistrations for Hyperion VNIR only from the observed data. The spectral property of band 41 was obtained by estimating the distortion of the oxygen absorption line in
the spectrum image. The detected band-to-band misregistrations for all subscenes revealed
their spatial property. We used cubic spline interpolation to correct the spectral signatures.
An evaluation of our proposed method demonstrated that the Hyperion VNIR characteristics were estimated and modified with 0.01-pixel accuracy. In addition, consistent propertiesof the sensor obtained from various scenes showed the robustness of the proposed method
to diﬀerent scenes. These results suggest that the accuracy of the proposed method estimating the Hyperion VNIR postlaunch characteristics is comparable to that of the prelaunchmeasurements, which enables the precise and robust onboard calibration of hyperspectral
sensors.
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List of Figure Captions
Fig. 1. (a)Original band 31, (b) MNF band 1 obtained from bands 8 to 57 for a desert scene
in Chile, (c) spectral profile of the scene, (d) weighting factors that contribute to MNF1.
Fig. 2. Spectral distortion of the oxygen absorption line detected by subpixel image registration based on (a) NCC method and (b) PC method.
Fig. 3. Comparison of prelaunch and estimated smile properties.
Fig. 4. Radiance diﬀerence between bands 40 and 42 for (a) the original data and the data
modified using (b) the prelaunch smile property and the properties estimated by (c) NCC
method and (d) PC method.
Fig. 5. Profiles of radiance diﬀerence between bands 40 and 42. The original data and thedata modified using the prelaunch smile property and the properties estimated by NCC
method and PC method are shown.
Fig. 6. Keystone property estimated by PC method for six cross-track numbers.
Fig. 7. Average keystone property estimated (a) before and (b) after correction.
Fig. 8. Spatial misregistration between bands 13 and 50 estimated from various spatially andtemporally diﬀerent scenes.

11

Table 1. Quadratic coeﬃcient, x -coordinate of axis of smile eﬀect, and RSS ofquadratic fitting.
Location
Acquisition Date
USA/Goldfield
2002/03/04
USA/Goldfield
2004/03/25
Chile
2002/10/01
Chile
2002/12/25
Average
Prelauncha(×10−6 ) x0 (pixel)−9.78285.90−9.89785.34−9.38284.86−9.70584.27−9.691
85.09−9.63878.19

12RSS(×10−3 )
7.674
6.821
8.795
4.047

Radiance (DN)

3000
2500
2000

Weighting Factor

(a)

(c)

3500

(d)

0.02
0.01
0
-0.01
-0.02
10

(b)

20 30 40 50
Band Number

Wavelength Shift (pixel)

Fig. 1. (a)Original band 31, (b) MNF band 1 obtained from bands 8 to 57 fora desert scene in Chile, (c) spectral profile of the scene, (d) weighting factorsthat contribute to MNF1.

0.05
0
-0.05
-0.1
-0.15
-0.2 (a)
-0.25
0

(b)
100
200 0
100
Cross-track Number (pixels)

200

Fig. 2. Spectral distortion of the oxygen absorption line detected by subpixel
image registration based on (a) NCC method and (b) PC method.

13

Wavelength Shift (pixel)

0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25

Prelaunch
NCC
PC
0
100
200
Cross-track Number (pixels)

Fig. 3. Comparison of prelaunch and estimated smile properties.

(a)

(b)

(c)

(d)

Fig. 4. Radiance diﬀerence between bands 40 and 42 for (a) the original data
and the data modified using (b) the prelaunch smile property and the properties estimated by (c) NCC method and (d) PC method.

14

Radiance Difference (DN)

100
0
-100
-200
Original
Prelaunch
NCC
PC

-300
-400
-500

0
100
200
Cross-track Number (pixels)

Spatial Shift (pixel)

Fig. 5. Profiles of radiance diﬀerence between bands 40 and 42. The original data and the data modified using the prelaunch smile property and theproperties estimated by NCC method and PC method are shown.

0.2
0.1
0
-0.1
-0.2

0.2
0.1
0
-0.1
-0.2

x = 16x = 56x = 96slope = -2.76×10 -3

slope = -2.81×10 -3

slope = -2.81×10 -3

x = 176x = 136slope = -2.83×10 -3

2040

x = 216

slope = -2.81×10 -3

60

20

40

slope = -2.82×10 -3

60

20

40

60

Band Number

Fig. 6. Keystone property estimated by PC method for six cross-track numbers.

15

Spatial Shift (pixel)

0.3
(a)

0.2

(b)

0.1
0
-0.1
-0.2
-0.3

10 20 30 40 50 60 10 20 30 40 50 60
Band Number

0.3
0.2
0.1
0
-0.1

01/10/02
02/03/04
02/05/04
02/07/08
02/07/31
02/10/01
02/10/09
02/12/25
03/01/31
03/02/23
04/03/25

Spatial Shift (pixel)

Fig. 7. Average keystone property estimated (a) before and (b) after correction.

Date

Fig. 8. Spatial misregistration between bands 13 and 50 estimated from various
spatially and temporally diﬀerent scenes.

16